SOLUTION: Prove that. (11/24) =< int(sqrt(1-x^2))dx =< (11/24)*(sqrt(4/3)),on [0,1/2] ,int means integral

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Question 1117377: Prove that. (11/24) =< int(sqrt(1-x^2))dx =< (11/24)*(sqrt(4/3)),on [0,1/2] ,int means integral
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!




You are comparing the size of three numbers. Since the smallest difference between any two of them is in the second decimal digit, it is sufficient to show their approximate values to two decimal places.

I'll leave the decimal representation of the two extreme values in your capable hands. You can operate a calculator as well as I can. This leaves us with the task of evaluating:



First thing is to find the antiderivative:



This is best solved by trigonometric substitution. Let . Then and , hence



Substitute:



Use the reduction formula:



for





Use the original substitutions:



Hence








The rest is just calculator work.

John

My calculator said it, I believe it, that settles it